Reduction of order

Reduction of order


A linear homogeneous second order differential equation is of the form:

a(x)y+b(x)y+c(x)y=0a(x) y''+b(x) y'+c(x)y=0

There will be two solution to the above differential equation:

y1(x)=f(x)y_1 (x)=f(x)
y2(x)=g(x)y_2 (x)=g(x)

And all general solutions will be of the form:

y(x)=c1f(x)+c2g(x)y(x)=c_1 f(x)+c_2 g(x)

The method of Reduction of order assumes that one solution is already know (i.e. we know that y1(x)=f(x)y_1 (x)=f(x) is a solution the above differential equation) and uses a specific method to find another solution.


We will use interchangeably v(x)=vv(x)=v, and other shorthand notation of that form.

Steps to solve Reduction of Order problems:
1) Assume that your second solution is of the form y2(x)=v(x)f(x)y_2 (x)=v(x)f(x), where y1(x)=f(x)y_1 (x)=f(x) is the solution we know.
2) Find y2y_2', and y2y_2''
3) Insert y2,y2y_2, y_2', and y2y_2'' into a(x)y+b(x)y+c(x)y=0a(x) y''+b(x) y'+c(x)y=0
4) Get rid of the solution with y1(x)=f(x)y_1 (x)=f(x), and rearrange your equation to isolate vv' and vv''
5) Substitute w=vw=v', and w=vw'=v''
6) Solve for ww in the new first order differential equation
7) Substitute w=vw=v' back in and then solve for vv
8) The second solution to the differential equation is y2(x)=v(x)f(x)y_2 (x)=v(x)f(x), with the vv found from the previous steps.
  • Introduction
    What is the Reduction of Order Method?

    A quick recap of the steps in the Reduction of Order Method

  • 1.
    Using Reduction of Order
    Find the general solution to the following homogeneous linear second order differential equation:

    Where it is known that y1(x)=c~1y_1 (x)=\tilde{c}_1 is a solution
    Where c~1\tilde{c}_1 is a constant.

  • 2.
    Find the particular solution to the following homogeneous linear second order differential equation:
    x2yxy+y=0x^2 y''-xy'+y=0

    Where it is known that y1(x)=xy_1 (x)=x is a solution. The initial values are y(1)=2y(1)=2, and y(1)=1y' (1)=1